On the power of quantum fingerprinting

In the simultaneous message model, two parties holding <i>n</i>-bit integers <i>x,y</i> send messages to a third party, the <i>referee</i>, enabling him to compute a boolean function <i>f(x,y)</i>. Buhrman et al [3] proved the remarkable result that, when <i>f</i> is the equality function, the referee can solve this problem by comparing short "quantum fingerprints" sent by the two parties, i.e., there exists a quantum protocol using only <i>O</i>(log <i>n</i>) bits. This is in contrast to the well-known classical case for which Ω(<i>n</i>^1/2) bits are provably necessary for the same problem even with randomization. In this paper we show that short quantum fingerprints can be used to solve the problem for a much larger class of functions. Let <i>R</i>^{<??par line>,<i>pub</i>}(<i>f</i>) denote the number of bits needed in the classical case, assuming in addition a common sequence of random bits is known to all parties (the <i>public coin</i> model). We prove that, if <i>R</i>^{<??par line>,<i>pub</i>}(<i>f</i>)=<i>O</i>(1), then there exists a quantum protocol for <i>f</i> using only <i>O</i>(log <i>n</i>) bits. As an application we show that <i>O</i>(log <i>n</i>) quantum bits suffice for the bounded Hamming distance function, defined by <i>f(x,y)</i>=1 if and only if <i>x</i> and <i>y</i> have a constant Hamming distance <i>d</i> or less.